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    "# 随机梯度下降\n",
    ":label:`sec_sgd`\n",
    "\n",
    "在前面的章节中，我们一直在训练过程中使用随机梯度下降，但没有解释它为什么起作用。为了澄清这一点，我们刚在 :numref:`sec_gd`中描述了梯度下降的基本原则。本节继续更详细地说明*随机梯度下降*（stochastic gradient descent）。\n"
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   "source": [
    "%matplotlib inline\n",
    "import math\n",
    "import torch\n",
    "from d2l import torch as d2l"
   ]
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    "## 随机梯度更新\n",
    "\n",
    "在深度学习中，目标函数通常是训练数据集中每个样本的损失函数的平均值。给定$n$个样本的训练数据集，我们假设$f_i(\\mathbf{x})$是关于索引$i$的训练样本的损失函数，其中$\\mathbf{x}$是参数向量。然后我们得到目标函数\n",
    "\n",
    "$$f(\\mathbf{x}) = \\frac{1}{n} \\sum_{i = 1}^n f_i(\\mathbf{x}).$$\n",
    "\n",
    "$\\mathbf{x}$的目标函数的梯度计算为\n",
    "\n",
    "$$\\nabla f(\\mathbf{x}) = \\frac{1}{n} \\sum_{i = 1}^n \\nabla f_i(\\mathbf{x}).$$\n",
    "\n",
    "如果使用梯度下降法，则每个自变量迭代的计算代价为$\\mathcal{O}(n)$，它随$n$线性增长。因此，当训练数据集较大时，每次迭代的梯度下降计算代价将较高。\n",
    "\n",
    "随机梯度下降（SGD）可降低每次迭代时的计算代价。在随机梯度下降的每次迭代中，我们对数据样本随机均匀采样一个索引$i$，其中$i\\in\\{1,\\ldots, n\\}$，并计算梯度$\\nabla f_i(\\mathbf{x})$以更新$\\mathbf{x}$：\n",
    "\n",
    "$$\\mathbf{x} \\leftarrow \\mathbf{x} - \\eta \\nabla f_i(\\mathbf{x}),$$\n",
    "\n",
    "其中$\\eta$是学习率。我们可以看到，每次迭代的计算代价从梯度下降的$\\mathcal{O}(n)$降至常数$\\mathcal{O}(1)$。此外，我们要强调，随机梯度$\\nabla f_i(\\mathbf{x})$是对完整梯度$\\nabla f(\\mathbf{x})$的无偏估计，因为\n",
    "\n",
    "$$\\mathbb{E}_i \\nabla f_i(\\mathbf{x}) = \\frac{1}{n} \\sum_{i = 1}^n \\nabla f_i(\\mathbf{x}) = \\nabla f(\\mathbf{x}).$$\n",
    "\n",
    "这意味着，平均而言，随机梯度是对梯度的良好估计。\n",
    "\n",
    "现在，我们将把它与梯度下降进行比较，方法是向梯度添加均值为0、方差为1的随机噪声，以模拟随机梯度下降。\n"
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    "def f(x1, x2):  # 目标函数\n",
    "    return x1 ** 2 + 2 * x2 ** 2\n",
    "\n",
    "def f_grad(x1, x2):  # 目标函数的梯度\n",
    "    return 2 * x1, 4 * x2"
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    "def sgd(x1, x2, s1, s2, f_grad):\n",
    "    g1, g2 = f_grad(x1, x2)\n",
    "    # 模拟有噪声的梯度\n",
    "    g1 += torch.normal(0.0, 1, (1,))\n",
    "    g2 += torch.normal(0.0, 1, (1,))\n",
    "    eta_t = eta * lr()\n",
    "    return (x1 - eta_t * g1, x2 - eta_t * g2, 0, 0)"
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      "epoch 50, x1: 0.225454, x2: -0.057039\n"
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      "  ary = asanyarray(ary)\n",
      "/home/d2l-worker/miniconda3/envs/d2l-zh-release-1/lib/python3.9/site-packages/torch/functional.py:478: UserWarning: torch.meshgrid: in an upcoming release, it will be required to pass the indexing argument. (Triggered internally at  ../aten/src/ATen/native/TensorShape.cpp:2895.)\n",
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    "def constant_lr():\n",
    "    return 1\n",
    "\n",
    "eta = 0.1\n",
    "lr = constant_lr  # 常数学习速度\n",
    "d2l.show_trace_2d(f, d2l.train_2d(sgd, steps=50, f_grad=f_grad))"
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    "正如我们所看到的，随机梯度下降中变量的轨迹比我们在 :numref:`sec_gd`中观察到的梯度下降中观察到的轨迹嘈杂得多。这是由于梯度的随机性质。也就是说，即使我们接近最小值，我们仍然受到通过$\\eta \\nabla f_i(\\mathbf{x})$的瞬间梯度所注入的不确定性的影响。即使经过50次迭代，质量仍然不那么好。更糟糕的是，经过额外的步骤，它不会得到改善。这给我们留下了唯一的选择：改变学习率$\\eta$。但是，如果我们选择的学习率太小，我们一开始就不会取得任何有意义的进展。另一方面，如果我们选择的学习率太大，我们将无法获得一个好的解决方案，如上所示。解决这些相互冲突的目标的唯一方法是在优化过程中*动态*降低学习率。\n",
    "\n",
    "这也是在`sgd`步长函数中添加学习率函数`lr`的原因。在上面的示例中，学习率调度的任何功能都处于休眠状态，因为我们将相关的`lr`函数设置为常量。\n",
    "\n",
    "## 动态学习率\n",
    "\n",
    "用与时间相关的学习率$\\eta(t)$取代$\\eta$增加了控制优化算法收敛的复杂性。特别是，我们需要弄清$\\eta$的衰减速度。如果太快，我们将过早停止优化。如果减少的太慢，我们会在优化上浪费太多时间。以下是随着时间推移调整$\\eta$时使用的一些基本策略（稍后我们将讨论更高级的策略）：\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    \\eta(t) & = \\eta_i \\text{ if } t_i \\leq t \\leq t_{i+1}  && \\text{分段常数} \\\\\n",
    "    \\eta(t) & = \\eta_0 \\cdot e^{-\\lambda t} && \\text{指数衰减} \\\\\n",
    "    \\eta(t) & = \\eta_0 \\cdot (\\beta t + 1)^{-\\alpha} && \\text{多项式衰减}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "在第一个*分段常数*（piecewise constant）场景中，我们会降低学习率，例如，每当优化进度停顿时。这是训练深度网络的常见策略。或者，我们可以通过*指数衰减*（exponential decay）来更积极地减低它。不幸的是，这往往会导致算法收敛之前过早停止。一个受欢迎的选择是$\\alpha = 0.5$的*多项式衰减*（polynomial decay）。在凸优化的情况下，有许多证据表明这种速率表现良好。\n",
    "\n",
    "让我们看看指数衰减在实践中是什么样子。\n"
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   "source": [
    "def exponential_lr():\n",
    "    # 在函数外部定义，而在内部更新的全局变量\n",
    "    global t\n",
    "    t += 1\n",
    "    return math.exp(-0.1 * t)\n",
    "\n",
    "t = 1\n",
    "lr = exponential_lr\n",
    "d2l.show_trace_2d(f, d2l.train_2d(sgd, steps=1000, f_grad=f_grad))"
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   "source": [
    "正如预期的那样，参数的方差大大减少。但是，这是以未能收敛到最优解$\\mathbf{x} = (0, 0)$为代价的。即使经过1000个迭代步骤，我们仍然离最优解很远。事实上，该算法根本无法收敛。另一方面，如果我们使用多项式衰减，其中学习率随迭代次数的平方根倒数衰减，那么仅在50次迭代之后，收敛就会更好。\n"
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      "epoch 50, x1: 0.052108, x2: 0.104767\n"
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   "source": [
    "def polynomial_lr():\n",
    "    # 在函数外部定义，而在内部更新的全局变量\n",
    "    global t\n",
    "    t += 1\n",
    "    return (1 + 0.1 * t) ** (-0.5)\n",
    "\n",
    "t = 1\n",
    "lr = polynomial_lr\n",
    "d2l.show_trace_2d(f, d2l.train_2d(sgd, steps=50, f_grad=f_grad))"
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    "关于如何设置学习率，还有更多的选择。例如，我们可以从较小的学习率开始，然后使其迅速上涨，再让它降低，尽管这会更慢。我们甚至可以在较小和较大的学习率之间切换。现在，让我们专注于可以进行全面理论分析的学习率计划，即凸环境下的学习率。对一般的非凸问题，很难获得有意义的收敛保证，因为总的来说，最大限度地减少非线性非凸问题是NP困难的。有关的研究调查，请参阅例如2015年Tibshirani的优秀[讲义笔记](https://www.stat.cmu.edu/~ryantibs/convexopt-F15/lectures/26-nonconvex.pdf)。\n",
    "\n",
    "## 凸目标的收敛性分析\n",
    "\n",
    "以下对凸目标函数的随机梯度下降的收敛性分析是可选读的，主要用于传达对问题的更多直觉。我们只限于最简单的证明之一 :cite:`Nesterov.Vial.2000`。存在着明显更先进的证明技术，例如，当目标函数表现特别好时。\n",
    "\n",
    "假设所有$\\boldsymbol{\\xi}$的目标函数$f(\\boldsymbol{\\xi}, \\mathbf{x})$在$\\mathbf{x}$中都是凸的。更具体地说，我们考虑随机梯度下降更新：\n",
    "\n",
    "$$\\mathbf{x}_{t+1} = \\mathbf{x}_{t} - \\eta_t \\partial_\\mathbf{x} f(\\boldsymbol{\\xi}_t, \\mathbf{x}),$$\n",
    "\n",
    "其中$f(\\boldsymbol{\\xi}_t, \\mathbf{x})$是训练样本$f(\\boldsymbol{\\xi}_t, \\mathbf{x})$的目标函数：$\\boldsymbol{\\xi}_t$从第$t$步的某个分布中提取，$\\mathbf{x}$是模型参数。用\n",
    "\n",
    "$$R(\\mathbf{x}) = E_{\\boldsymbol{\\xi}}[f(\\boldsymbol{\\xi}, \\mathbf{x})]$$\n",
    "\n",
    "表示期望风险，$R^*$表示对于$\\mathbf{x}$的最低风险。最后让$\\mathbf{x}^*$表示最小值（我们假设它存在于定义$\\mathbf{x}$的域中）。在这种情况下，我们可以跟踪时间$t$处的当前参数$\\mathbf{x}_t$和风险最小化器$\\mathbf{x}^*$之间的距离，看看它是否随着时间的推移而改善：\n",
    "\n",
    "$$\\begin{aligned}    &\\|\\mathbf{x}_{t+1} - \\mathbf{x}^*\\|^2 \\\\ =& \\|\\mathbf{x}_{t} - \\eta_t \\partial_\\mathbf{x} f(\\boldsymbol{\\xi}_t, \\mathbf{x}) - \\mathbf{x}^*\\|^2 \\\\    =& \\|\\mathbf{x}_{t} - \\mathbf{x}^*\\|^2 + \\eta_t^2 \\|\\partial_\\mathbf{x} f(\\boldsymbol{\\xi}_t, \\mathbf{x})\\|^2 - 2 \\eta_t    \\left\\langle \\mathbf{x}_t - \\mathbf{x}^*, \\partial_\\mathbf{x} f(\\boldsymbol{\\xi}_t, \\mathbf{x})\\right\\rangle.   \\end{aligned}$$\n",
    ":eqlabel:`eq_sgd-xt+1-xstar`\n",
    "\n",
    "我们假设随机梯度$\\partial_\\mathbf{x} f(\\boldsymbol{\\xi}_t, \\mathbf{x})$的$L_2$范数受到某个常数$L$的限制，因此我们有\n",
    "\n",
    "$$\\eta_t^2 \\|\\partial_\\mathbf{x} f(\\boldsymbol{\\xi}_t, \\mathbf{x})\\|^2 \\leq \\eta_t^2 L^2.$$\n",
    ":eqlabel:`eq_sgd-L`\n",
    "\n",
    "我们最感兴趣的是$\\mathbf{x}_t$和$\\mathbf{x}^*$之间的距离如何变化的*期望*。事实上，对于任何具体的步骤序列，距离可能会增加，这取决于我们遇到的$\\boldsymbol{\\xi}_t$。因此我们需要点积的边界。因为对于任何凸函数$f$，所有$\\mathbf{x}$和$\\mathbf{y}$都满足$f(\\mathbf{y}) \\geq f(\\mathbf{x}) + \\langle f'(\\mathbf{x}), \\mathbf{y} - \\mathbf{x} \\rangle$，按凸性我们有\n",
    "\n",
    "$$f(\\boldsymbol{\\xi}_t, \\mathbf{x}^*) \\geq f(\\boldsymbol{\\xi}_t, \\mathbf{x}_t) + \\left\\langle \\mathbf{x}^* - \\mathbf{x}_t, \\partial_{\\mathbf{x}} f(\\boldsymbol{\\xi}_t, \\mathbf{x}_t) \\right\\rangle.$$\n",
    ":eqlabel:`eq_sgd-f-xi-xstar`\n",
    "\n",
    "将不等式 :eqref:`eq_sgd-L`和 :eqref:`eq_sgd-f-xi-xstar`代入 :eqref:`eq_sgd-xt+1-xstar`我们在时间$t+1$时获得参数之间距离的边界，如下所示：\n",
    "\n",
    "$$\\|\\mathbf{x}_{t} - \\mathbf{x}^*\\|^2 - \\|\\mathbf{x}_{t+1} - \\mathbf{x}^*\\|^2 \\geq 2 \\eta_t (f(\\boldsymbol{\\xi}_t, \\mathbf{x}_t) - f(\\boldsymbol{\\xi}_t, \\mathbf{x}^*)) - \\eta_t^2 L^2.$$\n",
    ":eqlabel:`eqref_sgd-xt-diff`\n",
    "\n",
    "这意味着，只要当前损失和最优损失之间的差异超过$\\eta_t L^2/2$，我们就会取得进展。由于这种差异必然会收敛到零，因此学习率$\\eta_t$也需要*消失*。\n",
    "\n",
    "接下来，我们根据 :eqref:`eqref_sgd-xt-diff`取期望。得到\n",
    "\n",
    "$$E\\left[\\|\\mathbf{x}_{t} - \\mathbf{x}^*\\|^2\\right] - E\\left[\\|\\mathbf{x}_{t+1} - \\mathbf{x}^*\\|^2\\right] \\geq 2 \\eta_t [E[R(\\mathbf{x}_t)] - R^*] -  \\eta_t^2 L^2.$$\n",
    "\n",
    "最后一步是对$t \\in \\{1, \\ldots, T\\}$的不等式求和。在求和过程中抵消中间项，然后舍去低阶项，可以得到\n",
    "\n",
    "$$\\|\\mathbf{x}_1 - \\mathbf{x}^*\\|^2 \\geq 2 \\left (\\sum_{t=1}^T   \\eta_t \\right) [E[R(\\mathbf{x}_t)] - R^*] - L^2 \\sum_{t=1}^T \\eta_t^2.$$\n",
    ":eqlabel:`eq_sgd-x1-xstar`\n",
    "\n",
    "请注意，我们利用了给定的$\\mathbf{x}_1$，因而可以去掉期望。最后定义\n",
    "\n",
    "$$\\bar{\\mathbf{x}} \\stackrel{\\mathrm{def}}{=} \\frac{\\sum_{t=1}^T \\eta_t \\mathbf{x}_t}{\\sum_{t=1}^T \\eta_t}.$$\n",
    "\n",
    "因为有\n",
    "\n",
    "$$E\\left(\\frac{\\sum_{t=1}^T \\eta_t R(\\mathbf{x}_t)}{\\sum_{t=1}^T \\eta_t}\\right) = \\frac{\\sum_{t=1}^T \\eta_t E[R(\\mathbf{x}_t)]}{\\sum_{t=1}^T \\eta_t} = E[R(\\mathbf{x}_t)],$$\n",
    "\n",
    "根据詹森不等式（令 :eqref:`eq_jensens-inequality`中$i=t$，$\\alpha_i = \\eta_t/\\sum_{t=1}^T \\eta_t$）和$R$的凸性使其满足的$E[R(\\mathbf{x}_t)] \\geq E[R(\\bar{\\mathbf{x}})]$，因此，\n",
    "\n",
    "$$\\sum_{t=1}^T \\eta_t E[R(\\mathbf{x}_t)] \\geq \\sum_{t=1}^T \\eta_t  E\\left[R(\\bar{\\mathbf{x}})\\right].$$\n",
    "\n",
    "将其代入不等式 :eqref:`eq_sgd-x1-xstar`得到边界\n",
    "\n",
    "$$\n",
    "\\left[E[\\bar{\\mathbf{x}}]\\right] - R^* \\leq \\frac{r^2 + L^2 \\sum_{t=1}^T \\eta_t^2}{2 \\sum_{t=1}^T \\eta_t},\n",
    "$$\n",
    "\n",
    "其中$r^2 \\stackrel{\\mathrm{def}}{=} \\|\\mathbf{x}_1 - \\mathbf{x}^*\\|^2$是初始选择参数与最终结果之间距离的边界。简而言之，收敛速度取决于随机梯度标准的限制方式（$L$）以及初始参数值与最优结果的距离（$r$）。请注意，边界由$\\bar{\\mathbf{x}}$而不是$\\mathbf{x}_T$表示。因为$\\bar{\\mathbf{x}}$是优化路径的平滑版本。只要知道$r, L$和$T$，我们就可以选择学习率$\\eta = r/(L \\sqrt{T})$。这个就是上界$rL/\\sqrt{T}$。也就是说，我们将按照速度$\\mathcal{O}(1/\\sqrt{T})$收敛到最优解。\n",
    "\n",
    "## 随机梯度和有限样本\n",
    "\n",
    "到目前为止，在谈论随机梯度下降时，我们进行得有点快而松散。我们假设从分布$p(x, y)$中采样得到样本$x_i$（通常带有标签$y_i$），并且用它来以某种方式更新模型参数。特别是，对于有限的样本数量，我们仅仅讨论了由某些允许我们在其上执行随机梯度下降的函数$\\delta_{x_i}$和$\\delta_{y_i}$组成的离散分布$p(x, y) = \\frac{1}{n} \\sum_{i=1}^n \\delta_{x_i}(x) \\delta_{y_i}(y)$。\n",
    "\n",
    "但是，这不是我们真正做的。在本节的简单示例中，我们只是将噪声添加到其他非随机梯度上，也就是说，我们假装有成对的$(x_i, y_i)$。事实证明，这种做法在这里是合理的（有关详细讨论，请参阅练习）。更麻烦的是，在以前的所有讨论中，我们显然没有这样做。相反，我们遍历了所有实例*恰好一次*。要了解为什么这更可取，可以反向考虑一下，即我们*有替换地*从离散分布中采样$n$个观测值。随机选择一个元素$i$的概率是$1/n$。因此选择它*至少*一次就是\n",
    "\n",
    "$$P(\\mathrm{choose~} i) = 1 - P(\\mathrm{omit~} i) = 1 - (1-1/n)^n \\approx 1-e^{-1} \\approx 0.63.$$\n",
    "\n",
    "类似的推理表明，挑选一些样本（即训练示例）*恰好一次*的概率是\n",
    "\n",
    "$${n \\choose 1} \\frac{1}{n} \\left(1-\\frac{1}{n}\\right)^{n-1} = \\frac{n}{n-1} \\left(1-\\frac{1}{n}\\right)^{n} \\approx e^{-1} \\approx 0.37.$$\n",
    "\n",
    "这导致与*无替换*采样相比，方差增加并且数据效率降低。因此，在实践中我们执行后者（这是本书中的默认选择）。最后一点注意，重复采用训练数据集的时候，会以*不同的*随机顺序遍历它。\n",
    "\n",
    "## 小结\n",
    "\n",
    "* 对于凸问题，我们可以证明，对于广泛的学习率选择，随机梯度下降将收敛到最优解。\n",
    "* 对于深度学习而言，情况通常并非如此。但是，对凸问题的分析使我们能够深入了解如何进行优化，即逐步降低学习率，尽管不是太快。\n",
    "* 如果学习率太小或太大，就会出现问题。实际上，通常只有经过多次实验后才能找到合适的学习率。\n",
    "* 当训练数据集中有更多样本时，计算梯度下降的每次迭代的代价更高，因此在这些情况下，首选随机梯度下降。\n",
    "* 随机梯度下降的最优性保证在非凸情况下一般不可用，因为需要检查的局部最小值的数量可能是指数级的。\n",
    "\n",
    "## 练习\n",
    "\n",
    "1. 尝试不同的随机梯度下降学习率计划和不同的迭代次数进行实验。特别是，根据迭代次数的函数来绘制与最优解$(0, 0)$的距离。\n",
    "1. 证明对于函数$f(x_1, x_2) = x_1^2 + 2 x_2^2$而言，向梯度添加正态噪声等同于最小化损失函数$f(\\mathbf{x}, \\mathbf{w}) = (x_1 - w_1)^2 + 2 (x_2 - w_2)^2$，其中$\\mathbf{x}$是从正态分布中提取的。\n",
    "1. 从$\\{(x_1, y_1), \\ldots, (x_n, y_n)\\}$分别使用替换方法以及不替换方法进行采样时，比较随机梯度下降的收敛性。\n",
    "1. 如果某些梯度（或者更确切地说与之相关的某些坐标）始终比所有其他梯度都大，将如何更改随机梯度下降求解器？\n",
    "1. 假设$f(x) = x^2 (1 + \\sin x)$。$f$有多少局部最小值？请试着改变$f$以尽量减少它需要评估所有局部最小值的方式。\n"
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    "[Discussions](https://discuss.d2l.ai/t/3838)\n"
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